Social conditions and the Gompertz rate of ageing

Taking Gompertz Seriously

Jon AnsonYishai Friedlander

Deparment of Social WorkBen-Gurion University of the Negev

84105 Beer Sheva, Israel

Complexity in social systems: from data to models, Cergy-Pontoise, France, June 2013

Funding: ISF 677/11

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The Segmented Mortality CurveFrance, total population, 1913

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The Gompertz Model

• Samuel Gompertz (1825): Adult mortality increases exponentially with age

(x) = atbx

with t the mortality risk at age t and x the number of years past t

• Gompertz argued for t = 25. In practice, initial checks suggest we use t = 50

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Corollaries: Life table functions

2. Average yearsLived between t and x

1. Probability of Surviving x years

3. Density distribution

4. Modal age at death

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Criteria for goodness of fit

1. Probability of surviving from age 50 to age 95

2. Partial life expectancy over 45 years, between age 50 and 95

3. Modal age at death in density distribution

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Example: Two populations, at high and low mortality

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Gompertz lines at ages 50 to 95

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Fitted survivorship curves: l'50 = 1

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Density curves and modal ages at death

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Data I: A historical sample

• Sampled 108 male and female life tables from the Human Mortality Database (3,774 pairs)

• No two tables from the same year• Same country at least 25 years apart• Countries with historical long series over

represented

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Fitting mx: ages 50 to 95• 3-stage fitting process

– x = x – 50 (modelling years past age 50 – Fit log(mx) = a1 + x•log(b1) – Use a1 and b1 as starting points, fit

• mx = a2b2x (non-linear model)

– Use a2 and b2 as starting points, fit• xp50 =

– Use a3 and b3 for further analysis

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Reproducing partial life expectancy, ages 50 to 95

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Reproducing p(surviving) from age 50 to 95

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Reproducing the modal age at death

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Conclusions Stage I• At ages 50 to 95 (mature adult mortality) the

Gompertz model:– Reproduces partial life expectancy – Reproduces the details of the mortality distribution

(survivorship, modal age) but not perfectly– There is a marginal difference in the reproduction

beween male and female curves. For a given observed value:• p(surviving): Male > Female• Mode: Female > Male

• Question: which is more reliable, the data or the model?

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Dependence of b on a

• Large relative variation in mortality rate at age 50

• Little variation at age 95• Implies: the lower is a, the

the steeper the increase

Sample mortality slopes for Sample of values of a

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a and b : One parameter or two?

Question: what explains the residual variation in b?= delayed or premature adult mortality

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Data II: WHO contemporary• Slope (b) not determined uniquely by prior

mortality (a). Look at social conditions• 193 pairs of contemporary life tables for

2009, source: WHO. – Note: quality mixed, some data based; some

data + model; some model based. • Social data from UN Human Development

Index; Economist Intelligence Unit, etc.

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The social meaning of b• The human life span is effectively limited to

about 110 years, by which age all societies reach a similar level of mortality

• If mortality at mid adulthood (50) is low, mortality rates will increase more rapidly to attain this maximum – hence the strong negative relation between a and b

• All else being equal, advantageous social conditions will hold back the increase in the mortality rate (i. e. reduce b)

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Predicting b from social dataVariable Males Females Males FemalesIntercept 1.0943

(0.00101)1.0913

(0.00133)1.0948

(0.00121)1.0921

(0.00158)

Log(a) -0.0282(0.00190)

-0.0284(0.00100)

-0.0291(0.000723)

-0.0289(0.00105)

Log(GDP) -0.00219(0.000671)

-0.00427(0.000851)

-0.00303(0.000741)

-0.00483(0.000968)

Democracy -0.00129(0.000277)

-0.00190(0.000322)

-0.00166(0.000322)

-0.00236(0.000388)

Log(Gini) -0.00708(0.00236)

-0.00722(0.00283)

N Countries 132 104R² 0.9939 0.9945

Multi-level model with sex|Country variation, variables centred at median

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Interpreting social effects• The major determinant of the slope is the level of

mortality at younger ages (a)• The rate of increase for females is less steep than

for males• There is a considerable amount of missing data,

particularly concerning income and income distributions, mostly for poorer countries

• At lower levels of average income the mortality slope is steeper than at higher levels

• The more democratic a country, the less steep the mortality slope

• The greater the inequality, the less steep the mortality slope!!! (Survival effect?)

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Summary I • The human mortality curve can be broken down

into a number of log-linear segments, each of which can be fitted by a Gompertz model

mx = abx

• The Gompertz model above age 50 adequately reproduces the general level of mortality at these ages (partial life expectancy), but differs in detail from the published life table

• We cannot tell if these differences are due to the inadequacies of the model, or shortcomings in the data on which the life tables are based

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Summary II• The rate of increase in mortality (slope) above age

50 is heavily dependent on the level of mortality at age 50: the lower the mortality, the steeper the slope

• Given the starting level (a)– Female slopes are less steep than male slopes– High national income reduces the slope– Democratic government reduces the slope– Inequality reduces the slope!!!– The effects of wealth and democracy are greater for

females than for males

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Conclusion• Even allowing for mortality at younger ages,

there are important variations in mortality levels and rates of increase in mature adulthood

• These differences are related to the level of wealth and forms of social, economic and political organisation

• The Gompertz model provides a useful shorthand for summarising and investigating these differences

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Jon Ansonanson@bgu.ac.il